The window fill rate in a periodic review inventory system with order crossover

Many periodic review inventory studies make the simplifying assumption that orders do not crossover, that is, that they are delivered in the same sequence as they were issued. In many real-life situations, however, long international shipping routes result with frequent crossovers. Accordingly, we investigate a periodic review inventory system with backlogged compound Poisson demand in which order may crossover. We develop an exact formula for the window fill rate, i.e. the probability for customers to receive service within their tolerable wait. Evaluating the window fill rate using the exact formula is very time consuming and therefore an efficient approximation formula that assumes that orders do not crossover is considered. Lead times of actual global supply lines demonstrate that this approximation results in considerable overstocking when customers' tolerable wait is high, or understocking when the tolerable wait is low. Replacing the actual lead times distribution with the effective lead times distribution improves the approximation formula's accuracy considerably. This result allows us to show how the window fill rate depends on the model parameters and make practical observations about the trade-off between the tolerable wait and stock levels needed to maintain a required level of window fill rate.